Immediately algebraically closed fields

We consider two overlapping classes of fields, IAC and VAC, which are defined using valuation theory but which do not involve a distinguished valuation. Rather, each class is defined by a condition that quantifies over all possible valuations on the field. In his thesis, Hong asked whether these two classes are equal (Hong, 2013, Question 5.6.8). In this paper, we give an example that negatively answers Hong's question. We also explore several situations in which the equivalence does hold with an additional assumption, including the case where every $K'\equiv K$ is IAC.Comment: 12 pages, based on results from a chapter of the author's thesis, under the supervision of Professor Deirdre Haskel

NIP\rm NIP, and NTP2{\rm NTP}_2 division rings of prime characteristic

Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain $\rm NIP$ valued fields of characteristic $p$ with Dickson's construction of cyclic algebras, we provide examples of noncommutative $\rm NIP$ division ring of characteristic $p$ and show that an $\rm NIP$ division ring of characteristic $p$ has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite $\rm NIP$ fields have no Artin-Schreier extension. The result extends to ${\rm NTP}_2$ division rings of characteristic $p$, using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern $\rm NIP$ or simple difference fields

Finite Undecidability in Fields II: PAC, PRC and PpC Fields

A field $K$ in a ring language $\mathcal{L}$ is finitely undecidable if \mbox{Cons}(\Sigma) is undecidable for every nonempty finite \Sigma \subseteq \mbox{Th}(K; \mathcal{L}). We adapt arguments originating with Cherlin-van den Dries-Macintyre/Ershov (for PAC fields), Haran (for PRC fields), and Efrat (for PpC fields) to prove all PAC, PRC, and (bounded) PpC fields are finitely undecidable. This work is drawn from the author's PhD thesis and is a sequel to arXiv:2210.12729.Comment: 24 page

Model Theory and Groups

The aim of the workshop was to discuss the connections between model theory and group theory. Main topics have been the interaction between geometric group theory and model theory, the study of the asymptotic behaviour of geometric properties on groups, and the model theoretic investigations of groups of finite Morley rank around the Cherlin-Zilber Conjecture

Henselianity in the language of rings

We consider four properties of a field K related to the existence of (de-finable) henselian valuations on K and on elementarily equivalent fields and study the implications between them. Surprisingly, the full pictures look very different in equichar- acteristic and mixed characteristic